Question

the graph has four even vertices.
choose the correct answer below.
a.
b
a c
d
b.
a b
d c
c.
a b
d c
d.
a b
d c
e. such a graph is not possible

Explanation:

Step1: Recall the definition of an even vertex

An even vertex is a vertex with an even - numbered degree (the number of edges incident to the vertex).

Step2: Analyze option B

• For option B:
• Degree of vertex \(A\): Count the number of edges connected to \(A\). There are \(3\) edges (to \(B\), \(C\), and \(D\)).
• Degree of vertex \(B\): Count the number of edges connected to \(B\). There are \(3\) edges (to \(A\), \(C\), and \(D\)).
• Degree of vertex \(C\): Count the number of edges connected to \(C\). There are \(3\) edges (to \(A\), \(B\), and \(D\)).
• Degree of vertex \(D\): Count the number of edges connected to \(D\). There are \(3\) edges (to \(A\), \(B\), and \(C\)).
• For option A:
• Degree of vertex \(A\): \(2\) (edges to \(B\) and \(D\)).
• Degree of vertex \(B\): \(3\) (edges to \(A\), \(C\), and \(D\)).
• Degree of vertex \(C\): \(2\) (edges to \(A\) and \(D\)).
• Degree of vertex \(D\): \(4\) (edges to \(A\), \(B\), and \(C\) (counting multiple edges correctly)).
• For option C:
• Degree of vertex \(A\): \(3\) (edges to \(C\) and \(D\) (counting multiple edges)).
• Degree of vertex \(B\): \(1\) (edge to \(C\)).
• Degree of vertex \(C\): \(4\) (edges to \(A\), \(B\), and \(D\) (counting multiple edges)).
• Degree of vertex \(D\): \(3\) (edges to \(A\) and \(C\) (counting multiple edges)).
• For option D:
• Degree of vertex \(A\): \(2\) (edges to \(B\) and \(D\)).
• Degree of vertex \(B\): \(3\) (edges to \(A\), \(C\), and multiple - edges).
• Degree of vertex \(C\): \(2\) (edges to \(B\) and \(D\)).
• Degree of vertex \(D\): \(3\) (edges to \(A\) and \(C\) (counting multiple edges)).

Answer:

B.